Numbers in a Beatty sequence which are orders only of cyclic, abelian or nilpotent groups

TL;DR

This paper investigates the distribution of cyclic, abelian, and nilpotent numbers within Beatty sequences, extending previous asymptotic results by adapting established methods to these sequences.

Contribution

It demonstrates that the counting functions for these special numbers in Beatty sequences follow asymptotic formulas similar to prior results, scaled by a factor of 1/lpha.

Findings

Asymptotic formulas for counting functions in Beatty sequences are established.

Distribution of these numbers in Beatty sequences differs from previous results by a factor of 1/lpha.

Results apply to irrational lpha of finite type and fixed eta.

Abstract

Let \(C(x)\), \(A(x)\), and \(N(x)\) denote the counting functions of cyclic, abelian, and nilpotent numbers not exceeding \(x\), respectively. Their asymptotic formulas have been established in recent work by Pollack and Just. In this paper, by adapting the methods of Pollack and Just, we study the distribution of these numbers in Beatty sequences \(\mathcal{B}_{\alpha,\beta} = ([\alpha n + \beta])_{n=1}^{\infty}\), where \(\alpha > 1\) is an irrational number of finite type and \(\beta\) is a fixed real number. We prove that the counting functions \(\#C^*(x)\), \(\#A^*(x)\), and \(\#N^*(x)\) for cyclic, abelian, and nilpotent numbers in Beatty sequences satisfy asymptotic formulas that differ from those of Pollack and Just only by a factor \(1/\alpha\).